A study on the moduli space of representations and the twisted Alexander invariant

本文的目的是阐明扭曲亚历山大不变量（TAI）作为基本群的SL（2，C）-表示的模空间上的函数的基本性质，并给出从TAI导出三维流形的几何信息的框架。结果如下。(1)给出了纽结无限序列的TAI的次数和模空间的定义方程，它是扭纽结的推广。作为一个应用，我们将这些纽结纤维化的一个充分必要条件描述为一类有限性定理。(2)对于环面纽结，我们给出了SL（2，C）-表示模空间上的函数TAI的一个显式公式和特征。

The purpose of this research was to clarify fundamental properties of the twisted Alexander invariant (TAI) as a function on the moduli space of SL(2,C)-representations of the fundamental group and to give a framework for deriving geometric information of 3-manifolds from the TAI. The results are as follows.(1) We gave explicit formulas of degrees of the TAI and the defining equation of the moduli space for an infinite sequence of knots which is a generalization of twist knots. As an application, we described a necessary and sufficient condition that these knots are fibered as a kind of finiteness theorems.(2) For torus knots, we gave an explicit formula and a characterization of the TAI as a function on the moduli space of SL(2,C)-representations.